In this work, we consider general symmetric norms, which generalize
\(\ell _p\)  norms. More formally, a norm norm \(l : \mathbb {R}^d \rightarrow \mathbb {R}\)  is symmetric if, for all \(x \in \mathbb {R}^d\)  and every \(d \times d\)  permutation matrix \(P\) , it satisfies \(l(Px) = l(x)\)  and also \(l(|x|) = l(x)\) , where \(|x|\)  is the coordinate-wise absolute value of \(x\) 
(for a broader introduction, see [1]} Chapter IV).
Important special cases of symmetric norms are \(\ell _p\)  norms and Orlicz norms [2]}, which naturally arise in harmonic analysis and model data with sub-gaussian properties.
Other practical examples of symmetric norms include
top-\(k\)  norms, max-mix of \(\ell _p\)  norms, sum-mix of \(\ell _p\)  norms, the \(k\) -support norm [3]} and the box-norm [4]}.
